Structural and functional analysis of supply chain reliability in the presence of demand fluctuations Alexander N. Pavlova,b, Dmitry A. Pavlova, Valentin N. Vorotyagina , Alexander B. Umarova a Mozhaisky Military Space Academy, Zhdanovskaya str., 13, St. Petersburg, 197198, Russia b St. Petersburg Federal Research Center of the Russian Academy of Sciences, V.O. 14 line, 39, St. Petersburg, 199178, Russia Abstract The following article presents an approach to modelling, evaluating and analysing the structural and functional reliability and survivability of supply chains in conditions of fluctuating demand. The article proposes the concept of a parametric genome of the structure of complex multi-mode objects for calculating integral indicators of the structural and functional reliability of the supply chain with dynamic consumer orders. Keywords Structural and functional reliability, supply chain, parametric genome. 1 Introduction1 coming years it will be possible to talk about a paradigm shift in supply chain optimization: a In the process of implementing supply chain transition from minimizing costs to ensuring a management (SCM) in practice, managers have balance of efficiency and survivability. In this faced with the problem of adapting to customers’ regard, a promising direction for future research unplanned orders and individual technological is the development of models and formulas for and economic requirements. How and with what calculating structural and functional indicators of methods and technologies is it possible to assess reliability and survivability of supply chains, the reliability and stability of the supply chain in taking into account fluctuations in demand. the event of more or less serious deviations and violations, fluctuations? So, finance losses 2 The concept of a parametric derived due to not received orders, fines and genome of the structure. Integral penalties in certain supply chains reach 15% of the annual turnover [1]. In modern SCM, the indicators of the structural and final consumer of products has become the most functional reliability of the supply important link. Therefore, one of the key reasons chain for improving the reliability and survivability of supply chains is caused by the necessity to meet These indicators should be used as an their requirements taking into account changing additional factors of economic efficiency to demand. This issue is reflected in the following characterize the target (functional) conflict "efficiency or reliability" from an objective point works by [2-11]. It should be noted that the current trend in understanding the efficiency of of view [1]. To analyse the properties of the supply chain is the design of such supply structural and functional reliability and chains that would be characterized by a high survivability of the supply chain, as well as for level of economic efficiency and the required the structural and functional synthesis of a level of survivability [12, 13]. It seems that in the system corresponding to a given level of structural reliability and survivability, it is necessary to introduce a quantitative Models and Methods for Researching Information System in Transport, Dec. 11-12, St.Petersburg, Russia measurement. As a rule, structural and functional EMAIL: Pavlov62@list.ru (A.N. Pavlov); Dpavlov239@mail.ru analysis of complex objects to which the supply (D.A. Pavlov); Vorotyagin@rambler.ru (V.N. Vorotyagin); Antropicier737@gmail.com (A.B. Umarov). chain belongs begins with the construction of ORCID: 0000-0002-6193-8882 (A.N. Pavlov); 0000-0002-9238- their functional integrity diagram [14, 15]. The 9505 (D.A. Pavlov); 0000-0003-1324-3231 (V.N. Vorotyagin); diagram of the functional integrity of any 0000-0003-3308-8806 (A.B. Umarov). ©️ 2020 Copyright for this paper by its authors. Use permitted under Creative complex object allows to represent graphically Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) logical conditions for the implementation of their 61 own functions by elements and subsystems, as successful functioning (1). well as the goals of modelling the logical ( P1 , P2 ,..., Pn , Pn1 ,..., Pn m , Q1 , Q2 ,..., Qn , Qn1 ,..., Qn m ), (1) conditions for the implementation of the system where Pi (Qi ), i  1,..., n is probability of uptime property under study, for example, reliability or (failure) of functional elements of the supply failure, safety or occurrence of an accident, the chain, and Pn i (Qn i  1  Pn i ), i  1,..., m can be creation of certain operation modes of the object, etc. The structure of the constructed circuit interpreted either as the probability of receipt includes functional elements which are actually (non-receipt) of an order by the consumer, or the the various technological operations, subsystems, relative size (intensity) from 0 to 1 receipt blocks, nodes, connections of various physical (absence) of a sales order. nature. In the most general case, the functional vertices of the functional integrity scheme reflect both the operability of certain functional elements (for the supply chain, these can be suppliers, manufacturing plants, warehouses, distributors, providers, etc.), and the need for the implementation of certain functions for example, customer orders). Figure 1 shows a diagram of the functional integrity of a certain adaptive supply chain, taking into account the structural and functional reserve. Apexes 1-10 reflect the performance of individual product suppliers, manufacturing plants, transport enterprises involved in meeting Figure 1. Supply chain functional integrity the needs of consumers, represented by vertices diagram 11-14. Vertices from 15 to 33 are fictitious and are used to describe the logical relationships of Let us denote the intensity of receipt of orders the functional elements of the supply chain. from customers of the supply chain through For example, apex 33 ensures the customer satisfaction that is a successful operation i  Pni , i  1,..., m . Further, based on the (achievement of the goal) of the supply chain. assumption that all functional elements of the So, if there is no need in customer orders despite supply chain are homogeneous in the probability the refusals of suppliers and manufacturers, the of no-failure operation ( P1  P2  ...  Pn  P) , goal of the supply chain will be achieved. It the probabilistic polynomial of the successful should be noted that orders from consumers in functioning of the supply chain (1) can be the supply chain differ in the nature and the transformed to the following form (2) intensity of their receipt. Firstly, certain orders ( P, 1 ,  2 ,...,  m )  0 (1 ,  2 ,...,  m )  1 (1 ,  2 ,...,  m ) P  (2) can be main or auxiliary. In other words, the  2 (1 ,  2 ,...,  m ) P2  ...   n (1 ,  2 ,...,  m ) Pn orders may be inconsistent (that is, they may be By analogy with the concept of the structure executed one at a time), and individual orders genome introduced in [15-18], we will call may come in at the same time as others. vector  (1 ,  2 ,...,  m )  (  0 (1 ,  2 ,...,  m ), Secondly, orders may consist of different fractions of the arrival time at a given time 1 (1 ,  2 ,...,  m ),  2 (1 ,  2 ,...,  m ),...,  n (1 ,  2 ,...,  m ))Т interval or may have different values of the the parametric genome of the structure. arrival probability at a given interval, that is, they Using the parametric genome of the supply can have a deterministic or random dynamic chain structure, it is possible to calculate nature. Therefore, it is required to analyse and to estimates of the structural and functional evaluate indicators of the structural and reliability of the supply chain, depending on the functional reliability of the supply chain in the parameters 1 ,  2 ,...,  m of the intensity of conditions of joint and separate receipt of customer orders. dynamic customer orders. So, in the case of a probabilistic description Using the program complex of logical and of the failure-free operation of functional probabilistic modelling "Arbiter" [14], we obtain elements for a homogeneous structure (the same for the scheme of functional integrity of the probability of failure-free operation of functional supply chain a probabilistic polynomial of its elements), the function of the successful 62 functioning of the supply chain, represented by its functional elements, one can use [15-17] a the polynomial ( P, 1 ,  2 ,...,  m )  fuzzy integral as far as possible   0 (1 ,  2 ,...,  m )  1 (1 ,  2 ,...,  m ) P   2 (1 ,  2 ,...,  m ) P 2  Fhomogposs (  (1 ,  2 ,...,  m ))  ...   n (1 ,  2 ,...,  m ) P n , changes its values in  sup min{R(  , 1 ,  2 ,...,  m ), g (  )}  (5) [0,1] the interval [0,1]. Moreover, the closer the function graph to line ( P, 1 ,  2 ,...,  m )  1 ,  sup min{ , G ({ R(  , 1 ,  2 ,...,  m )   })}  [0,1] the higher the structural and functional reliability For this indicator, it is necessary to determine of the supply chain. Therefore, as an indicator of the measure of possibility G and, if possible, its structural and functional reliability in this case, it distribution function g ( ) . is proposed to use 1 For monotone homogeneous structures, the Fhomog (  (1 ,  2 ,...,  m ))   ( P, 1 ,  2 ,...,  m )dP graph of the polynomial of the possibility of 0 failure-free operation R(  , 1 ,  2 ,...,  m ) has the Then, to calculate the indicator of the structural form shown in Figure 3. Here the polynomial and functional reliability of the supply chain, you R (  , 1 ,  2 ,...,  m )   0 (1 ,  2 ,...,  m )  can use the parametric genome of the structure  1 (1 ,  2 ,...,  m )    2 (1 ,  2 ,...,  m )  2  ... according to the following formula (3) 1 ...   n (1 ,  2 ,...,  m )  n Fhomog (  (1 ,  2 ,...,  m ))   ( P, 1 ,  2 ,...,  m )dP  is obtained from the polynomial ( P, 1 ,  2 ,...,  m ) by replacing the probability 0 n 1    i (1 ,  2 ,...,  m )  P of no-failure operation of functional elements i 1 with the possibility  of no-failure operation of i 0 or 1 functional elements of the supply chain. As a Fhomog (  (1 ,  2 ,...,  m ))   ( P, 1 ,  2 ,...,  m )dP  measure of opportunity, we will use 0 (3) G ({ R(  , 1 ,  2 ,...,  m )   })  1 1 1 T   (1 ,  2 ,...,  m )  (1, , ,..., ) .  G ( H )  sup A  sup {1  } , 2 3 n 1  A H  R (  ,1 , 2 ,..., m )  In the case of a heterogeneous structure (different probability of failure-free operation of where A is the Lebegue measure. Therefore, in functional elements), it can be used as an the case of monotone homogeneous structures, indicator of the structural and functional the distribution function of the measure of reliability of the supply chain possibility is g ( )  1   . Fheterog (  (1 ,  2 ,...,  m ))  1 1 G   ... ( P1 , P2 ,..., Pn , 1 ,  2 ,...,  m ) dP1 dP2 ...dPn 0 0 1.0  or using the parametric genome structure 0.8 formula (4) Fheterog (  (1 ,  2 ,...,  m ))  0.6 1 1 1 (4) 0.4 G(H )   (1 ,  2 ,...,  m )  (1, , 2 ,..., n )T . Fhomogposs 2 2 2 0.2 In the case, when performing functions by 0  R(,1,...,m) functional elements included in the structure of 0.2 1.0 0.2 0.4 0.6 0.8 0.4 the supply chain, it is not possible to identify a 0.6 well-defined stochastic regularity of failure-free 0.8 1 operation, then it is proposed to use a fuzzy- R(,1,...,m) possibility approach to describing the behavior of  H functional elements, which is based on the concept of space with a measure of possibility Figure 2. Graphic interpretation of finding the [15-17]. indicator of the possibility of failure-free So, as an indicator (5) of the structural and operation of homogeneous structures functional reliability of the supply chain with a Then, for the considered case of a fuzzy- fuzzy-possibility description of the behaviour of possibility description of the failure-free 63 operation of functional elements of the supply use the capabilities of the general logical- chain, the indicator of the possibility of failure- probabilistic method [14] and introduce some free operation of a monotonic homogeneous "weights" of orders that would take into account structure can be calculated by the formula (6) the above differences. The weighting factors are Fhomogposs ( (1 , 2 ,..., m ))  1  * , (6) proposed to be introduced as follows. The where μ* is a solution to the equation weighting factor is found as the ratio of the average total duration of the order receipt during  (1 ,  2 ,...,  m )  (1, * , *2 ,..., *n )T  1  * . the considered time interval of the supply chain For non-monotone homogeneous structures, operation to the value of this interval. the uptime polynomial R(  , 1 ,  2 ,...,  m ) either does not preserve "0" ( R(0, 1 ,  2 ,...,  m )  1) , or 3 Research of structural and does not preserve "1" ( R(1, 1 ,  2 ,...,  m )  0) . functional reliability of the supply The graphs of the polynomials of the failure- chain free operation are shown in Figure 3. Fragment of the polynomial of the circuit When R(1, 1 ,  2 ,...,  m )  0 , the measure of functional integrity of the circuit shown in possibility is Figure 2 has the following form G ({ R(  , 1 ,  2 ,...,  m )   })  G ( H  )  ( P, 1 ,  2 ,...,  m )  P 4 (1  P)6 (1   2 )(1   4 )   P 2 (1  P)3 (1  1 )(1   2 ) 3 (1   4 )  ...  sup A  sup { max   min }, A H  R (  ,1 , 2 ,..., m )  ...  4 P5 (1  1 )(1   2 ) 3 (1   4 ). To study the structural and functional reliability where  max  sup{ R(, 1 , 2 ,..., m )   } of the supply chain, we will use formulas (3), (4), and  min  inf{ R(, 1 , 2 ,..., m )   } . (5). In this case, we will assume that all orders When R(0,1,2 ,...,m )  1, the measure of can be fulfilled both individually and jointly. The calculation results for possibility is i {0, 0.2, 0.4, 0.6, 0.8,1}, i  1,..., 4, are shown G ({ R (  , 1 ,  2 ,...,  m )   })  G ( H  )  in Figure 4.  sup A  sup {1  (  max   min )}, A H  R (  ,1 , 2 ,..., m )  where  max  sup{ R(, 1 , 2 ,..., m )   } and  min  inf{ R(, 1 , 2 ,..., m )   } . The graphical interpretation of finding the indicator of the possibility of failure-free operation in this case is shown in Figure 3. G 1.0  G(H ) 0.8 0.6 0.4 Figure 4. Structural and functional reliability of Fhomogposs 0.2 the supply chain  0.2 0 0.8 R(,1,...,m) Variant №1 corresponds to the absence of 1.0 H 0.2 0.4 0.6 0.4 orders (1  0;  2  0;  3  0;  4  0) . Variants 0.6 0.8 from №2 to №6 reflect a gradual increase from 0.2 to 1.0 in the intensity of order 1 (the apex of R(,1,...,m)  the functional integrity diagram №11) with a step of 0.2. Variants from No. 7 to No. 11, from No. Figure 3. Graphical interpretation of finding the 12 to No. 16, from No. 17 to No. 21 correspond indicator of the possibility of failure-free to an increase in the intensity of orders 2, 3, 4 operation of non-monotonic homogeneous (apexes No. 12, No. 13, No. 14). structures Further options from No. 22 to No. 26, from To study the structural and functional No. 27 to No. 31, from No. 32 to No. 36 reflect reliability of the supply chain, it is advisable to 64 the structural and functional reliability of a and joint receipt of three orders (2, 3, 4; 1, 2, 3; gradual increase in the joint receipt of two orders 1, 2, 4; 1, 3, 4). Then there are variants from No. 1 and 2, 1 and 3, 1 and 4, respectively. Then the 57 to No. 60 - a gradual increase in the intensity options for the joint receipt of three orders are of uniform separate and joint receipt of four located (1, 2, 3; 1, 2, 4; 1, 3, 4; 2, 3, 4). And orders. finally, options from 57 to 61 - a joint receipt of It should be noted that with the joint receipt four orders. It should be noted that with the of several orders (two, three, four), the feasible individual receipt of orders, the structural and assessment of the reliability of a homogeneous functional reliability of the supply chain is higher structure increases only by 4  6% . The than with the joint receipt of these orders. In probabilistic assessment of the reliability of a addition, it can be seen from the graphs that the homogeneous or heterogeneous structure can structural and functional reliability has a increase for two orders by a maximum of 12% , piecewise linear relationship with changes in the for three and four orders - by a maximum of intensities of the receipt of individual orders. 14.5  15.2% . Moreover, the maximum value is Moreover, for a homogeneous supply chain, it is achieved with the uniform use of these orders. impossible to clearly identify the best option for the joint receipt of orders, in contrast to a 4 Conclusions heterogeneous system. Within the framework of this article, a In Figure 5 shows the changes in the values of methodology and technology for assessing the indicators of the structural and functional reliability and survivability of the supply chain in reliability of the supply chain during the the event of more or less serious fluctuations in transition from joint to separate receipt of orders. demand are proposed. The proposed approach to Let us comment on the results obtained. the study of the structural and functional reliability of the supply chain under the conditions of changing customer orders is based on the parametric genome of the structure. The analysis of the above results showed that when creating and designing a supply chain, it is necessary to take into account various options (joint-incompatible, equivalent-unequal, homogeneous-heterogeneous) of dynamic customer orders, which significantly affect the reliability and survivability of the supply chain. Acknowledgments Figure 5. Increments of values of indicators structural and functional reliability of the supply Research carried out on this topic was carried out with partial financial support from RFBR chain with separate receipt of orders grants (No. 17-29-07073, 18-07-01272, 18-08- The first 20 options reflect a single receipt of 01505, 19–08–00989, 20-08-01046), under the orders with a gradual increase in intensity. It budget theme 0073–2019–0004. should be noted that in this case, the values of the References indicators Fheterog ( (1 ,..., 4 )) , [1] Ivanov, D.A., Ivanova, M.A. (2015). 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